The size of the associated Weyl group.

The number of proper elements in the Weyl group of a finite Cartan type.

The number of linear extensions of a certain poset defined for an integer partition. The poset is constructed in David Speyer's answer to Matt Fayers' question [3]. The value at the partition $\lambda$ also counts cover-inclusive Dyck tilings of $\lambda\setminus\mu$, summed over all $\mu$, as noticed by Philippe Nadeau in a comment. This statistic arises in the homogeneous Garnir relations for the universal graded Specht modules for cyclotomic quiver Hecke algebras.

The size of the centralizer of any permutation of given cycle type. The centralizer (or commutant, equivalently normalizer) of an element $g$ of a group $G$ is the set of elements of $G$ that commute with $g$: $$C_g = \{h \in G : hgh^{-1} = g\}.$$ Its size thus depends only on the conjugacy class of $g$. The conjugacy classes of a permutation is determined by its cycle type, and the size of the centralizer of a permutation with cycle type $\lambda = (1^{a_1},2^{a_2},\dots)$ is $$|C| = \Pi j^{a_j} a_j!$$ For example, for any permutation with cycle type $\lambda = (3,2,2,1)$, $$|C| = (3^1 \cdot 1!)(2^2 \cdot 2!)(1^1 \cdot 1!) = 24.$$ There is exactly one permutation of the empty set, the identity, so the statistic on the empty partition is $1$.

The sum of the vertex degrees of a graph. This is clearly equal to twice the number of edges, and, incidentally, also equal to the trace of the Laplacian matrix of a graph. From this it follows that it is also the sum of the squares of the eigenvalues of the adjacency matrix of the graph. The Laplacian matrix is defined as $D-A$ where $D$ is the degree matrix (the diagonal matrix with the vertex degrees on the diagonal) and where $A$ is the adjacency matrix. See [1] for detailed definitions.

The product of the parts of an integer partition.

The number of non-isomorphic minors of a graph. A minor of a graph $G$ is a graph obtained from $G$ by repeatedly deleting or contracting edges, or removing isolated vertices. This statistic records the total number of (non-empty) non-isomorphic minors of a graph.

The number of edges of a graph.

The number of multiset partitions such that the multiplicities of elements are given by a partition. In particular, the value on the partition $(n)$ is the number of integer partitions of $n$, [[oeis:A000041]], whereas the value on the partition $(1^n)$ is the number of set partitions [[oeis:A006110]].

The length of a longest trail in a graph. A trail is a sequence of distinct edges, such that two consecutive edges share a vertex.